Mechanical properties of cellular structures, including the cell cytoskeleton, are increasingly used as biomarkers for disease diagnosis and fundamental studies in cell biology. Recent experiments suggest that the cell cytoskeleton and its permeating cytosol, can be described as a poroelastic (PE) material. Biot theory is the standard model used to describe PE materials. Yet, this theory does not account for the fluid viscous stress, which can lead to inaccurate predictions of the mechanics in the dilute filamentous network of the cytoskeleton. Here, we adopt a two-phase model that extends Biot theory by including the fluid viscous stresses in the fluid's momentum equation. We use generalized linear viscoelastic (VE) constitutive equations to describe the permeating fluid and the network stresses and assume a constant friction coefficient that couples the fluid and network displacement fields. As the first step in developing a computational framework for solving the resulting equations, we derive closed-form general solutions of the fluid and network displacement fields in spherical coordinates. To demonstrate the applicability of our results, we study the motion of a rigid sphere moving under a constant force inside a PE medium, composed of a linear elastic network and a Newtonian fluid. We find that themore »
A generalized Rayleigh-Plesset equation for ions with solvent fluctuations
We introduce a mathematical modeling framework for the conformational dynamics of charged molecules (i.e., solutes) in an aqueous solvent (i.e., water or salted water). The solvent is treated as an incompressible fluid, and its fluctuating motion is described by the Stokes equation with the Landau–Lifschitz stochastic stress. The motion of the solute-solvent interface (i.e., the dielectric boundary) is determined by the fluid velocity together with the balance of the viscous force,hydrostatic pressure, surface tension, solute-solvent van der Waals interaction force, and electrostatic force. The electrostatic interactions are described by the dielectric Poisson–Boltzmann theory.Within such a framework, we derive a generalized Rayleigh–Plesset equation, a nonlinear stochastic ordinary differential equation (SODE), for the radius of a spherical charged molecule, such as anion. The spherical average of the stochastic stress leads to a multiplicative noise. We design and test numerical methods for solving the SODE and use the equation, together with explicit solvent molecular dynamics simulations, to study the effective radius of a single ion. Potentially, our general modeling framework can be used to efficiently determine the solute-solvent interfacial structures and predict the free energies of more complex molecular systems.
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- SIAM journal on applied mathematics
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- National Science Foundation
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